According to the National Cancer Institute for women between

According to the National Cancer Institute, for women between 50 and 59, there is a 2.38% chance of being diagnosed with breast cancer. Screening mammography has a sensitivity of about 85% for women over 50 and a specificity of about 95%. That is, the false-negative rate is 15% and the falsepositive rate is 5%. If a woman over 50 has a mammogram and it comes back positive for breast cancer, what is the probability that she has the disease?

Solution

Let

D = has disease
P = positive
N = negative

Then

P(D|P) = P(D) P(P|D) / P(P)

As

P(P) = P(D) P(P|D) + P(D\') P(P|D\') = 0.0238*0.85 + (1-0.0238)*0.05

= 0.06904

Thus,

P(D|P) = P(D) P(P|D) / P(P) = 0.0238*0.85/0.06904

P(D|P) = 0.29301854 [ANSWER]